Problem 6 - Advanced Macroeconomics

Question

The Condorcet paradox. Suppose there are three voters, \(1,2,\) and \(3,\) and three possible policies, \(A, B,\) and \(C .\) Voter 1 's preference ordering is \(A, B, C ;\) voter \(2^{\prime} \mathrm{s}\) is \(\mathrm{B}, \mathrm{C}, \mathrm{A} ;\) and voter \(3^{\prime} \mathrm{s}\) is \(\mathrm{C}, \mathrm{A}, \mathrm{B} .\) Does any policy win a majority of votes in a two-way contest against each of the alternatives? Explain.

Step-by-Step Solution

Answer

No policy wins against every alternative; this is the Condorcet paradox.

Step 1: Understanding Voter Preferences

We have three voters and they have expressed their preferences over three policies, labeled A, B, and C. Voter 1 prefers A over B over C. Voter 2 prefers B over C over A. Lastly, Voter 3 prefers C over A over B.

Step 2: Pairwise Comparison: A vs B

Compare policies A and B based on the voters' preferences: - Voter 1 prefers A to B. - Voter 2 prefers B to A. - Voter 3 prefers C to A but indirectly implies preference for B since both alternatives are preferred over A. Thus, B wins over A with a 2 to 1 majority.

Step 3: Pairwise Comparison: A vs C

Compare policies A and C based on the voters' preferences: - Voter 1 prefers A to C. - Voter 2 prefers B over C, implying preference for C over A. - Voter 3 prefers C to A. Thus, C wins over A with a 2 to 1 majority.

Step 4: Pairwise Comparison: B vs C

Compare policies B and C based on the voters' preferences: - Voter 1 prefers A over B, implying preference for B over C. - Voter 2 prefers B to C. - Voter 3 prefers C to B. Thus, B wins over C with a 2 to 1 majority.

Step 5: Conclusion to Condorcet Paradox

From the pairwise comparisons, we concluded: - B wins against A. - C wins against A. - B wins against C. No policy wins a majority of votes against each of the alternatives in every pairwise contest, illustrating the Condorcet paradox.

Key Concepts

Voter Preferences

Voter preferences describe the order in which voters rank options according to their liking or priorities. In this problem, each voter ranks three policies: A, B, and C, based on personal preference. These preferences are represented as lists such as A > B > C, where the sign '>' denotes 'prefers over'. This allows us to extract important information about each voter's desired ranking.

- **Voter 1** ranks choices as: A > B > C, meaning they find A the most appealing, then B, and C the least. - **Voter 2** ranks choices as: B > C > A, showing a preference for B first, followed by C, finally preferring A the least. - **Voter 3** selects: C > A > B, where C is highest in their preferences, A next, and B is least liked.
Understanding voter preferences is crucial to evaluating how different policies fare against each other in hypothetical two-way contests.

Pairwise Comparison

A pairwise comparison checks which of two choices is preferred according to voters' preferences. This involves comparing two policies at a time to see which gets more support. From the given voter preferences, we conduct multiple comparisons of policy pairs.

### Comparing A and B - **Voter 1**: Prefers A to B. - **Voter 2**: Prefers B to A. - **Voter 3**: Prefers C to A, indicating B is preferred indirectly because B is better than the alternatives. - In this case, B wins over A (2:1).
### Comparing A and C - **Voter 1**: Prefers A to C. - **Voter 2**: Prefers B to A but implies C over A. - **Voter 3**: Prefers C to A. - Here, C wins over A (2:1).
### Comparing B and C - **Voter 1**: Prefers A over B; thus, B is preferred to C. - **Voter 2**: Prefers B to C. - **Voter 3**: Prefers C to B. - Consequently, B wins over C (2:1).
This comparison process helps determine which policy wins under majority voting.

Majority Voting

Majority voting decides the preferable option by seeing which choice has the most votes or preference among the voters when compared individually against other options. Each policy goes through pairwise comparisons to see if it can win against each of the other policies.

In our setup with voters and policies, these pairwise results show that: - **B** defeats **A** in a majority (2:1). - **C** wins over **A** with a majority (2:1). - **B** triumphs over **C** also with a majority (2:1).
Even though B wins most comparisons, it does not dominate all alternatives because each result doesn't follow transitive logic. In majority voting like this, no strong consistent winner exists.

Social Choice Theory

Social choice theory studies methods of collective decision-making, analyzing how individual preferences lead to a collective outcome. An important concept in this theory is the Condorcet Paradox, which arises when collective preferences are non-transitive, even if individual preferences are transitive.

This paradox occurs when no single option can consistently win when compared with every other alternative in individual contests. In our scenario: - **B** beats **A** and **C**, but **C** beats **A**, creating a circular situation. - This means there isn't a single policy that always wins against every alternative as expected.
The Condorcet Paradox highlights that majority voting does not always yield a clear winner, reflecting complexities in aggregating diverse individual preferences into a societal choice. Understanding this paradox helps in designing fairer voting systems that might, for example, employ methods to break ties or use additional rounds to decide a winner.